## X(8) (NAGEL POINT)

 Interactive Applet

You can move the points A, B and C (click on the point and drag it).
Press the keys “+” and “−” to zoom in or zoom out the visualization window and use the arrow keys to translate it.

You can also construct all centers related with this one (as described in ETC) using the “Run Macro Tool”. To do this, click on the icon , select the center name from the list and, then, click on the vertices A, B and C successively.

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Download all construction files and macros: tc.zip (10.1 Mb).
This applet was built with the free and multiplatform dynamic geometry software C.a.R..

 Information from Kimberling's Encyclopedia of Triangle Centers

Trilinears           (b + c - a)/a : (c + a - b)/b : (a + b - c)/c
= csc2(A/2) : csc2(B/2) : csc2(C/2)

Barycentrics    b + c - a : c + a - b : a + b - c

Let A'B'C' be the points in which the A-excircle meets BC, the B-excircle meets CA, and the C-excircle meets AB, respectively. The lines AA', BB', CC' concur in X(8). Another construction of A' is to start at A and trace around ABC half its perimeter, and similarly for B' and C'. Also, X(8) is the incenter of the anticomplementary triangle.

X(8) is the {X(69),X(75)}-harmonic conjugate of X(7).

X(8) = reflection of X(I) in X(J) for these (I,J): (1,10), (4,355), (20,40), (100,1145), (145,1), (149,80), (192,984), (390,9), (944,2), (962,4), (1320,11), (1482,5), (1483,140)

X(8) = isogonal conjugate of X(56)
X(8) = isotomic conjugate of X(7)
X(8) = cyclocevian conjugate of X(189)
X(8) = complement of X(145)
X(8) = anticomplement of X(1)
X(8) = anticomplementary conjugate of X(8)
X(8) = X(I)-Ceva conjugate of X(J) for these (I,J): (69,329), (72,2), (312,346), (314,312), (333,9)

X(8) = X(I)-cross conjugate of X(J) for these (I,J):
(1,280), (9,2), (10,318), (11,522), (55,281), (72,78), (200,346), (210,9), (219,345), (497,7), (521,100)

X(8) = cevapoint of X(I) and X(J) for these (I,J):
(1,40), (2,144), (6,197), (9,200), (10,72), (11,522), (55,219), (175,176)

X(8) = crosspoint of X(I) and X(J) for these (I,J): (75,312), (314,333)

X(8) = crosssum of X(I) and X(J) for these (I,J):
(1,978), (31,604), (57,1423), (667,1015), (1042,1410), (1400,1402)

X(8) = crossdifference of any two points on line X(649)X(854)
X(8) = X(1)-aleph conjugate of X(1050)
X(8) = X(I)-beth conjugate of X(J) for these (I,J): (8,1), (341,341), (643,3), (668,8), (1043,8)

This is a joint work of
Humberto José Bortolossi, Lis Ingrid Roque Lopes Custódio and Suely Machado Meireles Dias.

If you have questions or suggestions, please, contact us using the e-mail presented here.

Departamento de Matemática Aplicada -- Instituto de Matemática -- Universidade Federal Fluminense